Special Function Inverse Series Pairs

Abstract

Working with the various special functions of mathematical physics and applied mathematics we often encounter inverse relations of the type and , where 0, 1, 2,. Here , denote special polynomial functions, and , denote coefficients found by use of the orthogonal properties of and , or by skillful series manipulations. Typically and , the n-th Legendre polynomial. We give a collection of inverse series pairs of the type if and only if , each pair being based on some reasonably well-known special function. We also state and prove an interesting generalization of a theorem of Rainville in this form.